# Revista Matemática Iberoamericana

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**Volume 23, Issue 3, 2007, pp. 1067–1114**

**DOI: 10.4171/RMI/524**

Published online: 2007-12-31

Strong $A_{\infty}$-weights and scaling invariant Besov capacities

Șerban Costea^{[1]}(1) Helsinki University of Technology, Finland

This article studies strong $A_{\infty}$-weights and Besov capacities as well as their relationship to Hausdorff measures. It is shown that in the Euclidean space ${\mathbb{R}}^n$ with $n\ge 2$, whenever $n-1 < s \le n$, a function $u$ yields a strong $A_\infty$-weight of the form $w=e^{nu}$ if the distributional gradient $\nabla u$ has sufficiently small $||\cdot||_{{\mathcal L}^{s,n-s}}({\mathbb{R}}^n; {\mathbb{R}}^n)$-norm. Similarly, it is proved that if $2\le n < p < \infty$, then $w=e^{nu}$ is a strong $A_\infty$-weight whenever the Besov $B_p$-seminorm $[u]_{B_p({\mathbb{R}}^n)}$ of $u$ is sufficiently small. Lower estimates of the Besov $B_p$-capacities are obtained in terms of the Hausdorff content associated with gauge functions $h$ satisfying the condition $\int_0^1 h(t)^{p'-1} \frac{dt}{t} < \infty$.

*Keywords: *strong $A_{\infty}$-weights, Besov spaces, capacity

Costea Șerban: Strong $A_{\infty}$-weights and scaling invariant Besov capacities. *Rev. Mat. Iberoam.* 23 (2007), 1067-1114. doi: 10.4171/RMI/524